Theorem mteqand

Description: A modus tollens deduction for inequality. (Contributed by Steven Nguyen, 1-Jun-2023)

	Ref	Expression
Hypotheses	mteqand.1	⊢ ( 𝜑 → 𝐶 ≠ 𝐷 )
	mteqand.2	⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐶 = 𝐷 )
Assertion	mteqand	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		mteqand.1	⊢ ( 𝜑 → 𝐶 ≠ 𝐷 )
2		mteqand.2	⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐶 = 𝐷 )
3	1	neneqd	⊢ ( 𝜑 → ¬ 𝐶 = 𝐷 )
4	3 2	mtand	⊢ ( 𝜑 → ¬ 𝐴 = 𝐵 )
5	4	neqned	⊢ ( 𝜑 → 𝐴 ≠ 𝐵 )